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Covariance & Covariance Matrices

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Covariance

A statistical measure of the joint variability of two variables, indicating whether they increase or decrease together.

Covariance Matrices

Diagonals are just variance of one column/row. Off diagonals are the Covariance.

Example (Context of VAMP-2):

Setup

  • Rows (e.g., 100): The Batch (The different molecules/frames).
  • Columns (e.g., 5): The States (The probability of being in State 1, State 2… State 5).

So, for a batch of 100 frames and 5 states, you have two matrices of size 100 × 5.

The Cross-Covariance is found by matrix-multiplying Output 1 Transposed by Output 2:
This mathematically compares every column against every other column. The result is a small 5 × 5 matrix (if you have 5 states) that tells you the entire story of the system’s dynamics.

What the Resulting 5x5 Matrix Tells You

Every cell in this new matrix represents a specific relationship:

  • The Diagonals (Row 1, Col 1):
    • Question: “Did the molecules that started in State 1 (at time t) stay in State 1 (at time t+τ)?”
    • Meaning: This measures Metastability (How stable the state is).
  • The Off-Diagonals (Row 1, Col 2):
    • Question: “Did the molecules that started in State 1 switch to State 2?”
    • Meaning: This measures the Transition Probability (The kinetics).

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